Stiffer than its cracked section says

When half the section has given up is a question about a section: at a crack, the concrete below the neutral axis carries nothing, the steel carries the whole tension, and the second moment of what is left is a third of the gross.

Between the cracks it has not given up. The steel is bonded to the concrete along its whole length, so as soon as the steel tries to stretch it drags the surrounding concrete into tension with it. Halfway between two cracks the concrete has recovered most of the tension it lost, the steel strain has dropped, and the curvature there is much closer to the uncracked value than to the cracked one.

The beam’s deflection is an integral of curvature, so what it responds to is the average.

Which free body produced the number #

Take a length of beam containing several cracks and ask for the average curvature over it.

At a crack the curvature is $M/E_cI_2$, the fully cracked value. Midway between cracks it is close to $M/E_cI_1$, the uncracked one. In between it varies with the bond stress transferring force back into the concrete, and the average over the length is somewhere between the two — closer to the cracked value the higher the moment, because a higher moment means more cracks, closer together, with less uncracked length between them.

The standard interpolation is

$$\frac{1}{r} = \zeta\left(\frac{1}{r}\right)_{II} + (1-\zeta)\left(\frac{1}{r}\right)_{I}, \qquad \zeta = 1 - \beta\left(\frac{M_{cr}}{M}\right)^2$$

and the thing being interpolated is the curvature, not the second moment. Those are different operations — a weighted average of $1/I_1$ and $1/I_2$ is not the reciprocal of a weighted average of $I_1$ and $I_2$ — and the difference is largest exactly where the two bounds are furthest apart.

The consequence is worth stating in plain terms: a member’s stiffness is not a property of any of its sections. Every section along the beam is either cracked or not, and neither of them has the stiffness the beam behaves with.

Integrating the curvature, rather than choosing a stiffness #

There is a second refinement that matters more than it looks.

Only part of the beam has cracked. Under a uniform load the moment is largest at mid-span and zero at the supports, so a beam whose mid-span moment is 3.5 times the cracking moment is uncracked over the outer fifth of its span and progressively less cracked as the section approaches the ends.

Choosing a single “effective second moment” and using it for the whole beam ignores that. Integrating the curvature — evaluating $\zeta$ at every station and putting the result through the unit-load integral — does not, and the two differ by several per cent on an ordinary beam and by more on a lightly loaded one where the cracked region is short.

This is the same observation about where a deflection comes from, arriving from the other side: the deflection is dominated by the middle half of the span, so what matters most is how cracked the middle half is, and the ends contribute little whatever their state.

The number that decides it is the one nobody can measure #

$M_{cr}$ is the tensile strength times a section modulus, and everything above turns on it: it fixes where the curve leaves the uncracked line and it sits squared inside $\zeta$.

Concrete’s tensile strength has a coefficient of variation around twenty per cent, is not measured directly in any routine test, and is usually inferred from the compressive strength by a correlation. It is comfortably the least reliable number in the calculation.

What that does to the answer depends entirely on where the beam is working.

A beam designed to sit near its cracking moment has put its answer on the property with the widest scatter, which is an unusual place for a design to end up and a common one. It happens whenever a member is sized by something other than deflection — by cover, by fire, by a standard depth — and ends up lightly stressed.

The corollary is more useful than the warning. A beam well past cracking is insensitive to the tensile strength, so a heavily loaded member’s deflection is more predictable than a lightly loaded one’s, in the ratio of about ten to one.

Bond deteriorates, and the interpolation knows it #

$\beta$ is the bond factor, and it is 1.0 for a single short-term load and 0.5 for sustained or repeated loading.

The reason is physical. What produces tension stiffening is bond between the bar and the concrete around it, and cycling a load breaks that bond down over a lengthening zone either side of each crack — so less of the concrete is recruited, the average curvature moves towards the fully cracked one, and the beam softens.

On the beam here it is worth about four per cent, from 23.2 mm to 24.2, because the beam is already at $\zeta = 0.92$ and there is little tension stiffening left to lose. On a beam near cracking it is worth far more, for exactly the reason the previous section gives.

And it compounds with creep rather than adding to it. Creep multiplies the concrete’s compressive strain and takes the effective modulus down; the loss of bond moves the interpolation towards the cracked bound. The two act on different terms of the same expression, and a long-term deflection calculation that includes only the first is a common and substantial underestimate.

Why the answer sits so close to the cracked bound #

The interpolated deflection here is 23.2 mm against bounds of 8.3 and 25.2 — 88 per cent of the way to the cracked one. That is typical, and it is worth understanding why, because it changes what tension stiffening is for.

$\zeta$ contains $(M_{cr}/M)^2$, and squaring a fraction is unkind to it. At twice the cracking moment $\zeta$ is 0.75; at three times it is 0.89; at the 3.5 times this beam reaches it is 0.92. So a beam at any ordinary service load is nearly fully cracked in the interpolation’s terms, and the uncracked bound — which is a factor of three away — is doing almost nothing.

Tension stiffening is therefore not a way of recovering stiffness on a working beam. It is worth a few per cent there. What it is for is the region just above cracking, where it is worth a factor of two and where the fully cracked calculation is badly wrong — and the region just above cracking is where every lightly loaded member in a building sits.

That inverts the usual presentation. The effect is described as a refinement to a deflection calculation, and it is a refinement on the beams that are easy and the whole answer on the beams that are not.

Which of the two checks is deciding the member therefore decides how much of this essay applies to it. A beam governed by strength is well past its cracking moment and insensitive to everything here; a beam governed by deflection is often not, and is exactly where the interpolation is carrying the answer. That stiffness is not strength is the usual reading of the distinction, and this is its sharpest consequence: the members whose deflection matters most are the members whose deflection is hardest to compute.

Where it shows up as something other than a number #

Tension stiffening is usually met as a deflection correction. It has two other appearances that are the same phenomenon wearing different clothes.

Crack width. The width of a crack is the difference between the steel’s extension and the concrete’s over the length between cracks — which is exactly the quantity tension stiffening is about. A calculation of crack width and a calculation of deflection are the same integral with different weights, and a member with more tension stiffening has both a smaller deflection and narrower cracks.

Redistribution. A continuous beam’s moments depend on the relative stiffness of its regions, and the hogging region over a support cracks before the sagging region at mid-span. So the structure softens unevenly, sheds moment from the support to the span, and does it progressively as the load rises — a redistribution nobody chose, produced entirely by the order in which the parts of the member cracked.

The redistribution nobody chose is usually discussed as a plastic phenomenon at the ultimate limit state. Most of it happens at service load, for a stiffness reason rather than a strength one, and this essay is that reason: the hogging region reaches its cracking moment first, its $\zeta$ rises first, and the moments move away from it while every material on the member is still elastic.

What a designer actually does with it #

The arithmetic above is what a computer does. What a designer does is a span-to-depth ratio, and it is worth saying how the two are related, because the connection is not obvious.

A span-to-depth limit is a deemed-to-satisfy rule: a table of depths, adjusted for the reinforcement ratio and the concrete grade, calibrated so that a member inside it will not exceed span over 250 in the long term. Every quantity in this essay is buried inside that calibration — the interpolation, the bond factor, an assumed creep coefficient, an assumed proportion of permanent load, and an assumed tensile strength.

Which is why the rule is generous where this essay says the answer is uncertain and tight where it says it is not. A lightly reinforced member gets a much smaller allowable ratio than a heavily reinforced one, and that is not because it is weaker; it is because a low reinforcement ratio means a low cracking moment relative to the applied one, which is the sensitive region.

The rule also assumes a beam that has cracked. A member that is genuinely uncracked in service — a prestressed one, or a very lightly loaded one — is three times stiffer than the table allows for, and the table has no way of saying so.

Why the rule is written in depth at all is span to the fourth: deflection goes as the fourth power of the span and the reciprocal of the second moment, so a depth limit is the only single number that can stand in for a deflection calculation — and it is standing in for everything on this page.

A slab is the same argument with the numbers moved #

Everything above is a beam, and the member that most often has its deflection computed is a slab.

A slab is more sensitive to all of it, for one reason: its reinforcement ratio is far lower. A beam at one per cent steel has a cracked second moment about a third of its gross; a slab at 0.3 per cent has one nearer a fifth, so the two bounds are further apart and the interpolation is carrying more of the answer. And because a slab is shallow, its cracking moment is a larger fraction of the moment it carries — which puts it in exactly the region the sensitivity curve says is worst.

Two consequences follow that a beam does not have. A slab’s deflection calculation is far more sensitive to whether it cracked at all, so a lightly loaded slab that stayed uncracked is several times stiffer than the arithmetic predicts and one that cracked during construction is several times softer. And a slab spanning two ways cracks in one direction before the other, so it is anisotropic in stiffness at service load whatever it was designed to be.

The member the calculation is most often asked about is also the one where the two bounds are furthest apart. A slab that spans both ways and has cracked in one direction and not the other is sharing its load in a ratio nobody chose, and neither direction’s stiffness is the one it was designed with.

The beam does not come back down the curve it went up #

The curve in the first figure is drawn once, going up. What the member does when the load is removed is a different curve, and the difference decides the number a deflection limit is actually written about.

Most limits are on an increment: the movement after the finishes were installed, which is the imposed load’s contribution and not the total. The obvious way to get it is to read the curve twice and subtract — $\delta$ at total load less $\delta$ at dead load. That is right for the first application and wrong for every one after it.

Cracking does not undo. Take the load off and the member unloads along a path close to the cracked line rather than back down the interpolation curve, because the cracks that formed are still there and the concrete between them has already had its bond degraded. The steel pulls the cracks nearly shut and the section does not recover its uncracked stiffness, so the member returns to a dead-load deflection larger than the one it started at.

Three consequences follow, and the third is the one worth carrying.

The first application of full service load produces the largest increment. Everything after it is smaller, because the softening has already happened. So the damaging event for a finish is the first time the building is fully loaded — not the hundredth.

Two identical beams can have different answers. A beam that carried its full service load before the partitions went up — a stacked-out floor during fit-out, a construction load, a water test — presents the partitions with a member that has already cracked and already softened. A beam whose first full loading comes after the partitions are in presents them with the whole increment. Same beam, same load, different damage, and the difference is a date.

And it is why $\beta$ drops to 0.5. The bond factor’s reduction for repeated or sustained loading is not a separate phenomenon; it is this one, averaged. What the code is saying is that a member which has been round the loop is permanently nearer its cracked bound, and it applies the reduction because it has no way of knowing how many times.

None of which is in the calculation, and the input it needs is not a structural quantity at all. The question is when the member first saw its full load relative to when the thing that cracks was attached to it, and that is a construction programme rather than an analysis. It is the same input the settlement timing argument needs and the same one nobody records — with the difference that here the structure remembers, and a member that has been overloaded once is softer for the rest of its life.

Where the model stops #

The interpolation is calibrated, not derived. The exponent of two on $M_{cr}/M$ comes from fitting tests; the underlying bond mechanics gives something close to it over the useful range and nothing exact. The expression is a good empirical summary of a real phenomenon rather than a solution to it.

Shrinkage is left out entirely, and it is not small. Restrained shrinkage puts the concrete into tension before any load arrives, which lowers the effective cracking moment — sometimes to half its nominal value — and it curves the member before it is loaded. A great many deflections that are blamed on tension stiffening being overestimated are shrinkage that was left out.

The section is singly reinforced. Compression steel restrains creep and shrinkage substantially, and a beam with a top mat behaves differently in the long term by an amount this calculation has no term for.

And the load is monotonic. The first application of a load cracks the member; the second finds it already cracked and follows a different, softer path. A member’s deflection under a load it has carried before is larger than under the same load applied for the first time, and neither of the curves drawn is that second path.

What the pictures cannot show #

The curve in the first figure is smooth. What a real beam does at the cracking moment is jump: a crack forms, that section’s stiffness drops in an instant, and the deflection steps. What is drawn is the average of many such steps at many sections, which is what a beam with fifteen cracks in it does and not what one with two does.

Nor can any figure show the thing a site engineer would ask about, which is whether the beam has cracked. Flexural cracks in a service-load member are a tenth of a millimetre wide and under a finish. The difference between the two bounds on this page is a factor of three in stiffness, and there is no way to tell from outside which of them a particular beam is nearer.

The assumption the figure rests on #

Every number here assumes the beam cracked because of the load in the calculation.

That is the assumption most often wrong. Members crack from restrained early thermal contraction while the concrete is a few days old, from restrained drying shrinkage over the following months, from handling if they were precast, and from loads applied during construction that are nowhere in the design case — a stack of blocks, a wet slab above, a propping arrangement that put the member into reverse curvature for a week.

A member that cracked for one of those reasons has an effective $M_{cr}$ of nearly zero and behaves as the fully cracked section from the first day. That is the honest upper bound on the deflection of a concrete beam, and the distance between it and the calculation — 25.2 mm against 23.2 here, and 25.2 against 12 on a lightly loaded one — is the width of the answer rather than an error in it.

That tensile strength is the flaw that sets the strength rather than a material constant — a statistical statement about the largest defect in the volume being stressed, which is why it scatters, why it falls with member size, and why a calculation that depends on it is a calculation with a range rather than a value. And the commonest way a member reaches it is not by being loaded.